Graded identities of matrix algebras and the universal graded algebra

Authors:
Eli Aljadeff, Darrell Haile and Michael Natapov

Journal:
Trans. Amer. Math. Soc. **362** (2010), 3125-3147

MSC (2000):
Primary 16W50, 16R10, 16R50; Secondary 16S35, 16K20

Published electronically:
January 7, 2010

MathSciNet review:
2592949

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider fine group gradings on the algebra of by matrices over the complex numbers and the corresponding graded polynomial identities. Given a group and a fine -grading on , we show that the -ideal of graded identities is generated by a special type of identity, and, as a consequence, we solve the corresponding Specht problem for this case. Next we construct a universal algebra (depending on the group and the grading) in two different ways: one by means of polynomial identities and the other one by means of a generic two-cocycle (this parallels the classical constructions in the nongraded case). We show that a suitable central localization of is Azumaya over its center and moreover, its homomorphic images are precisely the -graded forms of . Finally, we consider the ring of central quotients of which is a central simple algebra over the field of quotients of the center of . Using earlier results of the authors we show that this is a division algebra if and only if the group is one of a very explicit (and short) list of nilpotent groups. It follows that for groups not on this list, one can find a nonidentity graded polynomial such that its power is a graded identity. We illustrate this phenomenon with an explicit example.

**1.**Eli Aljadeff and Darrell Haile,*Division algebras with a projective basis*, Israel J. Math.**121**(2001), 173–198. MR**1818387**, 10.1007/BF02802503**2.**Eli Aljadeff, Darrell Haile, and Michael Natapov,*Projective bases of division algebras and groups of central type*, Israel J. Math.**146**(2005), 317–335. MR**2151606**, 10.1007/BF02773539**3.**Eli Aljadeff, Darrell Haile, and Michael Natapov,*On fine gradings on central simple algebras*, Groups, rings and group rings, Lect. Notes Pure Appl. Math., vol. 248, Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 1–9. MR**2226177**, 10.1201/9781420010961.ch1**4.**E. Aljadeff, C. Kassel,

Polynomial identities and noncommutative versal torsors,*arXiv:0708.4108, Adv. Math.*(2008), doi:10.1016/j.aim.2008.03.014.**5.**E. Aljadeff, M. Natapov,

On the universal -graded central simple algebra,*Actas del XVI Coloquio Latinoamericano de Álgebra*(Colonia, Uruguay, 2005), 115-130,

Revista Matemática Iberoamericana, Madrid, 2007.**6.**Eli Aljadeff and Jack Sonn,*Projective Schur algebras of nilpotent type are Brauer equivalent to radical algebras*, J. Algebra**220**(1999), no. 2, 401–414. MR**1717348**, 10.1006/jabr.1999.7860**7.**Yuri Bahturin and Vesselin Drensky,*Graded polynomial identities of matrices*, Linear Algebra Appl.**357**(2002), 15–34. MR**1935223**, 10.1016/S0024-3795(02)00356-7**8.**Yu. A. Bahturin, S. K. Sehgal, and M. V. Zaicev,*Group gradings on associative algebras*, J. Algebra**241**(2001), no. 2, 677–698. MR**1843319**, 10.1006/jabr.2000.8643**9.**Yu. A. Bahturin and M. V. Zaicev,*Group gradings on matrix algebras*, Canad. Math. Bull.**45**(2002), no. 4, 499–508. Dedicated to Robert V. Moody. MR**1941224**, 10.4153/CMB-2002-051-x**10.**Frank DeMeyer and Edward Ingraham,*Separable algebras over commutative rings*, Lecture Notes in Mathematics, Vol. 181, Springer-Verlag, Berlin-New York, 1971. MR**0280479****11.**Frank R. DeMeyer and Gerald J. Janusz,*Finite groups with an irreducible representation of large degree*, Math. Z.**108**(1969), 145–153. MR**0237629****12.**Pavel Etingof and Shlomo Gelaki,*A method of construction of finite-dimensional triangular semisimple Hopf algebras*, Math. Res. Lett.**5**(1998), no. 4, 551–561. MR**1653340**, 10.4310/MRL.1998.v5.n4.a12**13.**Pavel Etingof and Shlomo Gelaki,*The classification of triangular semisimple and cosemisimple Hopf algebras over an algebraically closed field*, Internat. Math. Res. Notices**5**(2000), 223–234. MR**1747109**, 10.1155/S1073792800000131**14.**Robert B. Howlett and I. Martin Isaacs,*On groups of central type*, Math. Z.**179**(1982), no. 4, 555–569. MR**652860**, 10.1007/BF01215066**15.**I. Martin Isaacs,*Character theory of finite groups*, Dover Publications, Inc., New York, 1994. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423 (57 #417)]. MR**1280461****16.**Gregory Karpilovsky,*Group representations. Vol. 2*, North-Holland Mathematics Studies, vol. 177, North-Holland Publishing Co., Amsterdam, 1993. MR**1215935****17.**Michael Natapov,*Projective bases of division algebras and groups of central type. II*, Israel J. Math.**164**(2008), 61–73. MR**2391140**, 10.1007/s11856-008-0020-7**18.**I. Reiner,*Maximal orders*, London Mathematical Society Monographs. New Series, vol. 28, The Clarendon Press, Oxford University Press, Oxford, 2003. Corrected reprint of the 1975 original; With a foreword by M. J. Taylor. MR**1972204**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
16W50,
16R10,
16R50,
16S35,
16K20

Retrieve articles in all journals with MSC (2000): 16W50, 16R10, 16R50, 16S35, 16K20

Additional Information

**Eli Aljadeff**

Affiliation:
Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel

Email:
aljadeff@tx.technion.ac.il

**Darrell Haile**

Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405

Email:
haile@indiana.edu

**Michael Natapov**

Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405

Address at time of publication:
Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel

Email:
mnatapov@indiana.edu, natapov@tx.technion.ac.il

DOI:
http://dx.doi.org/10.1090/S0002-9947-10-04811-7

Received by editor(s):
October 26, 2007

Received by editor(s) in revised form:
April 22, 2008

Published electronically:
January 7, 2010

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.